COSMO · STUDY + TEACHING DECK

Number Theory

September 9, 2025

A question worth sharing

Modular Arithmetic

Remainders, cycles, and replacing a huge number with the small piece that matters.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use modulus, remainder, congruent to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

What is the last digit of $7^{2026}$ ? Find a method you could explain without calculating the power.

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

List the last digits of $7^{1}$ , $7^{2}$ , $7^{3}$ , ...
Mark where the pattern first repeats.
Decide which part of 2026 chooses the answer.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

modulus

the cycle size

remainder

the position kept

congruent

same position in that cycle

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

a\equiv r\pmod m,\quad b\equiv s\pmod m\quad\Longrightarrow\quad ab\equiv rs\pmod m

The idea

Remainders, cycles, and replacing a huge number with the small piece that matters.

What to remember

A remainder is compressed information, not a rough estimate.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

What day is 100 days after Tuesday?

Predict each next step

A week repeats every 7 days.
100 = 14 x 7 + 2.
Keep the remainder 2.
Move two days: Thursday.

Which step needs a reason, not just a calculation?

See the structure

A clock is a remainder machine

Different numbers can land at the same place in a cycle.

Read the representation

On a 12-hour clock, 15 and 3 occupy the same position, so 15 is congruent to 3 mod 12.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

A useful shortcut

Reduce after each operation.
Keep the modulus fixed.
Explain why the reduction preserves the question.

A dangerous shortcut

Guess a cycle from too few terms.
Mix mod 4 and mod 10.
Report a pattern without proving it repeats.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

Guess a cycle from too few terms.
Mix mod 4 and mod 10.
Report a pattern without proving it repeats.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Build the last-digit atlas

Your shared task

Pairs choose bases 2 through 9 and record each repeating last-digit cycle.

Explore

4 min - generate powers

Compare

5 min - compare cycle lengths

Defend

4 min - explain one pattern

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

Every base eventually repeats because only ten remainders are possible.
Cycle length, not exponent size, controls the lookup.
A short table can encode an enormous calculation.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Find the remainder when $2^{50}$ is divided by 7.

Predict each next step

Powers of 2 mod 7 are 2, 4, 1.
The cycle length is 3.
50 leaves remainder 2 mod 3.
Use the second cycle value: 4.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

The divisibility-by-9 test is place value collapsing.

Build the argument

10 is congruent to 1 mod 9.
Every power of 10 is also congruent to 1.
A number and its digit sum therefore have the same remainder mod 9.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Find the last two digits of $3^{20}$ . What modulus should you choose, and what cycle do you need?

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Calendars, checksums, cryptography, and computer clocks all keep only the remainder information relevant to the system.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

Why is $7^{2026}$ a cycle-position problem?

Question 2

When may remainders be multiplied instead of the original numbers?

Question 3

Explain the divisibility-by-9 test in one complete sentence.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

A remainder is compressed information, not a rough estimate.

Try the worksheet without a calculator, then write one explanation another student could reuse.

A good question belongs to everyone in the room.